A class of Stephani cosmological models as a prototype of a non-homogeneous universe is considered. The non-homogeneity can lead to accelerated evolution, which is now observed from the SNe Ia data. Three samples of type Ia supernovae obtained by Perlmutter et al, Tonry et al and Knop et al are taken into account. Different statistical methods (best fits as well as maximum likelihood method) to obtain estimation for the model parameters are used. The Stephani model is considered as an alternative to the ΛCDM model in the explanation of the present acceleration of the universe. The model explains the acceleration of the universe at the same level of accuracy as the ΛCDM model (χ2 statistics are comparable). From the best fit analysis it follows that the Stephani model is characterized by a higher value of density parameter Ωm0 than the ΛCDM model. It is also shown that the model is consistent with the location of CMB peaks.

Class of spherically symmetric Stephani cosmological models is examined in the context of evolution type. It is assumed that the equation of state at the symmetry center of the models is barotropic (p(t) = αρ(t)) and the function k(t) playing role of spatial curvature is proportional to Stephani version of the Friedmann-Robertson-Lemaitre-Walker scale factor R(t) (k(t) = βR(t)).Classification of cosmological models is performed depending on different values and signs of parameters α and β. It is shown that for β < 0 (hyperbolic geometry) dust-like (α = 0) cosmological model exhibits accelerated expansion at later stages of evolution.The Hubble and deceleration parameters are defined in the model and it is shown that the deceleration parameter decreases with the distance becoming negative for sufficiently distant galaxies.Redshift-magnitude relation m(z) is calculated and discussed in the context of SnIa observational data. It is noticed that the most distant supernovae of type Ia fit quite well to the relation m(z) calculated in the considered model (H 0 = 65 km/sMpc, Ω 0 ≤ 0.3) without introducing the cosmological constant.It is also shown that the age of the universe in the model is longer than in the Friedmann model corresponding to the same H 0 and Ω 0 parameters.PACS numbers: 98.62. Py, 98.80.Es, 98.80.Hw 1

The Friedman equation for the universe with arbitrary curvature (k = 0, ±1), filled with mutually noninteracting pressureless dust, radiation, cosmological constant, and strings is considered. We assume the string domination scenario for the evolution of the latter component. Moreover, we discuss the simplest possibility for the scaling of the string energy density: ̺ ∝ R −2 . For such models we write down the explicit solution of the Friedman equation. We realize that corresponding cosmological models do not essentially differ from those without strings. We find an analytic formula for the radial coordinate χ of a galaxy with a redshift z and express it in terms of astronomical parameters. This relation is then used for the derivation of the astrophysical formulas such as luminosity distance, angular diameter, and source counts, which may serve for testing the string-dominated universe. It seems that the most sensitive test, at least from the formal point of view, is the formula for the number of galaxies N(z) corresponding to a given value of the redshift. We show that the maximum of N(z) strongly depends on the density of strings, especially if the density is large enough to explain the Ω problem. Other tests are also proposed.

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